Smooth Algebra

In algebra, a commutative ''k''-algebra ''A'' is said to be 0-smooth if it satisfies the following lifting property: given a ''k''-algebra ''C'', an ideal ''N'' of ''C'' whose square is zero and a ''k''-algebra map $u: A \to C/N$, there exists a ''k''-algebra map $v: A \to C$ such that ''u'' is ''v'' followed by the canonical map. If there exists at most one such lifting ''v'', then ''A'' is said to be 0-unramified (or 0-neat). ''A'' is said to be 0-étale if it is 0-smooth and 0-unramified. The notion of 0-smoothness is also called formal smoothness.

A finitely generated ''k''-algebra ''A'' is 0-smooth over ''k'' if and only if Spec ''A'' is a smooth scheme over ''k''.

A separable algebraic field extension ''L'' of ''k'' is 0-étale over ''k''. The formal power series ring k ![t_1, \ldots, t_n!">_1,_\ldots,_t_n.html" ;"title="![t_1, \ldots, t_n">![t_1, \ldots, t_n!/math> is 0-smooth only when $\operatornamek = p > 0$ and $[k: k^p] < \infty$ (i.e., ''k'' has a finite p-basis, ''p''-basis.)

''I''-smooth

Let ''B'' be an ''A''-algebra and suppose ''B'' is given the ''I''-adic topology, ''I'' an ideal of ''B''. We say ''B'' is ''I''-smooth over ''A'' if it satisfies the lifting property: given an ''A''-algebra ''C'', an ideal ''N'' of ''C'' whose square is zero and an ''A''-algebra map $u: B \to C/N$ that is continuous when $C/N$ is given the discrete topology, there exists an ''A''-algebra map $v: B \to C$ such that ''u'' is ''v'' followed by the canonical map. As before, if there exists at most one such lift ''v'', then ''B'' is said to be ''I''-unramified over ''A'' (or ''I''-neat). ''B'' is said to be ''I''-étale if it is ''I''-smooth and ''I''-unramified. If ''I'' is the zero ideal and ''A'' is a field, these notions coincide with 0-smooth etc. as defined above.

A standard example is this: let ''A'' be a ring, B = A ![t_1, \ldots, t_n!">_1,_\ldots,_t_n.html" ;"title="![t_1, \ldots, t_n">![t_1, \ldots, t_n!/math> and $I = (t_1, \ldots, t_n).$ Then ''B'' is ''I''-smooth over ''A''.

Let ''A'' be a noetherian local ring">local